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MODEL PAPER 2025 — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMODEL PAPER 20251 Mark
Question
The value of $\alpha$ if the angle between $\vec{p}=2 \alpha^2 \hat{\imath}-3 \alpha \hat{\jmath}+\hat{k}$ and $\vec{q}=\hat{\imath}+\hat{\jmath}+\alpha \hat{k}$ is obtuse, is
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Answer

For obtuse angle, $\cos \theta<0=>\vec{p} \cdot \vec{q}<0$
$2 \alpha^2-3 \alpha+\alpha<0 \Rightarrow 2 \alpha^2-2 \alpha<0 \Rightarrow \alpha\in(0,1)$

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